Cholera Dynamics and El Niño-Southern Oscillation Mercedes Pascual, et al. Science 289, 1766 (2000); DOI: 10.1126/science.289.5485.1766
The following resources related to this article are available online at www.sciencemag.org (this information is current as of March 8, 2007 ):
Updated information and services, including high-resolution figures, can be found in the online version of this article at: http://www.sciencemag.org/cgi/content/full/289/5485/1766
This article cites 9 articles, 3 of which can be accessed for free: http://www.sciencemag.org/cgi/content/full/289/5485/1766#otherarticles
This article has been cited by 79 article(s) on the ISI Web of Science.
This article has been cited by 19 articles hosted by HighWire Press; see: http://www.sciencemag.org/cgi/content/full/289/5485/1766#otherarticles
This article appears in the following subject collections: on March 8, 2007 Medicine, Diseases http://www.sciencemag.org/cgi/collection/medicine
Information about obtaining reprints of this article or about obtaining permission to reproduce this article in whole or in part can be found at: http://www.sciencemag.org/about/permissions.dtl www.sciencemag.org Downloaded from
Science (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by the American Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. Copyright c 2000 by the American Association for the Advancement of Science; all rights reserved. The title SCIENCE is a registered trademark of AAAS. R EPORTS
Although some uncertainties exist about these cli- International Earth Science Information Network at million live within currently malarious areas that are matic responses (23), the medium-high scenario ftp://ftp.ciesin.org/pub/data/Grid_Pop_World. This predicted to become unsuitable by 2050, a net in- ϩ commonly forms the basis of current attempts to record of the 1994 human population density per crease of 23 million, or 0.84% on the 1994 baseline predict the impact of climate change on human square kilometer was turned into a raster image at population data. For the high scenario, the corre- health. Outputs of the medium-high scenario are 1/12° spatial resolution and was subsequently used sponding figures are 389 million, 414 million, and a net decrease of 25 million or –0.92%, respectively the average of four separate GCM runs and are to estimate the total human population within the (Fig. 1C). given as differences between the modeled present malarious areas shown in Fig. 1, A through C, allow- 21. Z. K. Ma and R. L. Redmond, Photogramm. Eng. and modeled future conditions; the high-scenario ing for the different land areas corresponding to outputs are scaled versions of the medium-high Remote Sensing 61, 435 (1995). pixels at different latitudes. Land pixels in the malaria 22. D. J. Rogers, S. I. Hay, M. J. Packer, Ann. Trop. Med. outputs (23). Following usual practice, the GCM map imagery were mapped onto their equivalent 6 differences were added to the observed 30-year Parasitol. 90, 225 (1996). by 6 grid in the population density imagery, from climatic means (after cubic-spline interpolation to 23. M. Hulme and G. J. Jenkins, “Climate change scenar- which population totals were extracted and summed. the same spatial resolution), to generate the pre- ios for the UK: Scientific report” (Climatic Research dicted future climate surfaces that were used in This method estimated a total global population of Unit, Norwich, UK, 1998). the present analysis. 5611 million people in 1994, of which 2727 million 24. We thank the Department for International Develop- ment (grant R6626 to D.J.R.) and the Wellcome Trust 20. The “Gridded Population of the World” unsmoothed lived within the predicted malarious areas of Fig. 1A. (S.E.R.) for financial support and G. B. White and S. I. population density data file created by the Socioeco- Under the medium-high scenario, 357 million people Hay for helpful comments. nomic Data and Applications Center at Columbia Uni- live within areas that are currently malaria-free but versity (Palisades, NY) was obtained from the Center for are predicted to become malarious by 2050, and 334 31 March 2000; accepted 22 June 2000
forcing of disease transmission (9). To investi- Cholera Dynamics and El gate the role of ENSO in light of this alternative explanation, we consider a nonlinear time series Nin˜o–Southern Oscillation approach that allows us to compare specific alternative hypotheses for the underlying fac- Mercedes Pascual,1* Xavier Rodo´,2 Stephen P. Ellner,3 tors in cholera dynamics. Because the null Rita Colwell,4 Menno J. Bouma5 (non-ENSO) hypothesis is a nonlinear interac- tion between seasonality and cholera dynamics, Analysis of a monthly 18-year cholera time series from Bangladesh shows that the use of standard linear time series models the temporal variability of cholera exhibits an interannual component at the would strongly bias the comparison in favor of on March 8, 2007 dominant frequency of El Nin˜o–Southern Oscillation (ENSO). Results from the ENSO alternative. nonlinear time series analysis support a role for both ENSO and previous disease Lacking information that could be used to levels in the dynamics of cholera. Cholera patterns are linked to the previously specify a valid mechanistic model for the described changes in the atmospheric circulation of south Asia and, consistent ENSO effect, we use time series models that with these changes, to regional temperature anomalies. are both nonlinear and nonparametric and are effective at modeling high-dimensional rela- Cholera remains a major public health problem bacterium that causes the disease, is now known tionships. The dynamics of a variable of in-
in many areas of the world, including Bang- to inhabit brackish waters and estuarine systems terest, Nt, a measure of cholera levels, are ladesh and India. A climate influence on cholera (2) and thus might be sensitive to climate pat- modeled with a nonlinear equation of the www.sciencemag.org has long been debated (1), and it has been terns. Here we examine the associations be- form suggested that ENSO, a major source of inter- tween cholera and ENSO and between cholera annual climate variability, drives the interannual and climate at interannual time scales, using an ϭ ͩ Nt ϩ Tp f Nt, Nt Ϫ , Nt Ϫ 2, ... Nt Ϫ ͑d Ϫ 1͒, variation of the disease (2, 3). For example, 18-year record from Bangladesh where the dis- cholera reappeared in Peru with the El Nin˜o ease is endemic. A nonlinear time series ap- 2 2 ͪ ϩ event of 1991–92 and seems to fluctuate season- proach allows us to consider different hypothe- sin t, cos t, E Ϫ e (1) 12 12 t f t ally in Bangladesh with sea surface temperature ses for the roles of environmental driving vari- Downloaded from (SST) in the Bay of Bengal (2, 4). Recent ables and the inherent disease dynamics in pro- where Tp is a prediction time, f is a nonlinear studies of time series for diarrhoeal diseases in ducing the interannual variability of cholera. function, and Et is the environmental forcing Peruvian children have shown an increase in The disease data consist of a monthly time under consideration (10, 11). The sin and cos
cases associated with warmer temperatures and series for cholera incidence between January functions implement a seasonal clock and et the 1997–98 El Nin˜o (5, 6). Vibrio cholerae, the 1980 and March 1998 in Dhaka, Bangladesh represents the IID random noise variables. The (Fig. 1A). Over the same time span, the month- parameters , f , and d denote, respectively, 1Center of Marine Biotechnology, University of Mary- ly SST anomaly in a region of the equatorial two different time lags and the number of time land Biotechnology Institute, 701 East Pratt Street, Pacific provides an index for ENSO (Fig. 1B). delay variables. Time delay coordinates are Suite 236, Columbus Center, Baltimore, MD 21202, The cholera time series displays the well- used in the model as surrogates for unobserved USA, and Biology Department, Woods Hole Oceano- graphic Institution, Woods Hole, MA 02543, USA. known seasonal variation of the disease—typ- variables influencing the endogenous dynamics 2Climate Research Group, PCB–University of Barce- ically described as bimodal, with a small peak of the disease, such as the fraction of suscepti- lona, and Department of Ecology, University of Bar- in the spring and a larger one in the fall or early ble individuals in the population (12, 13). The celona, 08028 Barcelona, Catalunya, Spain. 3Depart- winter—but also shows a multiyear modulation functional form of f is not specified in a rigid ment of Ecology and Evolutionary Biology, Cornell of the seasonal cycles. The interannual variabil- form. Instead, the shape of f is determined by University, Ithaca, NY 14853, USA. 4Center of Marine Biotechnology, University of Maryland Biotechnology ity of cholera cases has a dominant frequency of the data, using an objective model selection Institute, Baltimore, MD 21202, USA, and Department 1/3.7 years, as shown by singular spectrum criterion: generalized cross-validation (GCV) of Cell and Molecular Biology, University of Maryland, analysis (7, 8) (Fig. 2). The same dominant (14). We used the GCV criterion to compare 5 College Park, College Park, MD 20742, USA. Depart- frequency is found for the ENSO time series, models with and without seasonality and with ment of Infectious and Tropical Diseases, London School of Hygiene and Tropical Medicine, University which suggests that climate variability acts as a and without the environmental covariate Et of London, London WC1E 7HT, UK. driver in the dynamics of the disease (Fig. 2). (Table 1). The selected model is low-dimen- *To whom correspondence should be addressed. E- Alternatively, however, this low-frequency sional and incorporates both seasonality and mail: [email protected] variability could arise solely from the seasonal ENSO as external forcings (Fig. 3). The model
1766 8 SEPTEMBER 2000 VOL 289 SCIENCE www.sciencemag.org R EPORTS also accounts for the largest fraction of the compensatory type: Cholera incidence increas- ities of transmission through water in the envi- variance in the data. es monotonically with previous levels but with ronment. Overcompensatory density depen- Could the better fit (the higher r2)ofthe an ever decreasing slope. Compensatory densi- dence, a negative effect of previous levels on model incorporating ENSO simply result from ty dependence can result from a decrease in the current levels, has been observed in other in- the larger number of variables and parameters number of susceptibles in the population after a fectious diseases (16) but is not apparent here in that model? To address this question, we large number of cases and/or from the complex- for a time lag of 2 months. Although we cannot conducted parametric bootstrap tests for the significance of the improvement in fit (15). The result of these comparisions (Table 2) shows Fig. 1. (A) Time series of percentage of chol- that the addition of the ENSO index as a pre- era cases obtained by dictor variable is highly significant. But so is the ICDDR,B (Interna- the addition of previous disease levels. These tional Centre for Diar- results support a role for both extrinsic factors rhoeal Disease Research, (ENSO and seasonality) and intrinsic ones Bangladesh, in Dhaka) (previous disease levels) in the dynamics of from a systematic sam- ple of the patients vis- cholera. iting the facility. (B) A positive influence of previous disease lev- ENSO time series given els indicates a density-dependent effect in chol- by the index Nin˜o 3.4. era transmission. In the fall, when the largest values generally occur, this effect is of the
Table 1. Comparison of models using the GCV model selection criterion Vc. The model (defined by Eqs. 1 and 2) is fitted to the cholera time series after the data are square-root–transformed, which normalizes the residuals and stabilizes the vari- Fig. 2. Singular spectrum analysis applied to the ance. A time lag t of 2 months is used, based on ENSO index and to the cholera time series with on March 8, 2007 the well-known rule of thumb of choosing the lag seasonality removed. For both time series, a for which the autocorrelation function first crosses window length of 60 months was used. The 0.5. Thus, Eq. 1 takes the general form dominant eigenvalues for both data sets are given by a pair, indicating the existence of an ϭ ͩ -d Ϫ 1͒, underlying oscillatory component. The ei͑ءYt ϩ 2 f Yt , Yt Ϫ 2, ... Yt Ϫ 2 genspectrum of nonlinear signals typically ex- 2 2 ͪ ϩ hibit three distinct regions: first, a group of sin t, cos t, E Ϫ e (4) 12 12 t f t significant eigenvalues, used here to obtain the ϭ ͌ reconstructed components (A); then an inter- where Yt Nt. When seasonality is incorporated but not the ENSO index, low-dimensional models mediate slope; and finally the noise floor. Error www.sciencemag.org (d ϭ 1 and d ϭ 2) are selected (that is, they have the bars were calculated from the inverse of the lag-one AR coefficient with conservative weights smallest values of the cross-validation criterion Vc). The importance of seasonality is demonstrated by according to (23, 24). Projections of the signal comparing these models to their autonomous coun- onto the empirical orthogonal functions (EOFs) terparts. Models with an equivalent or larger number of the corresponding eigenvalues give the dif- of independent variables (for example, d ϭ 4 and ferent principal components (PCs). To recon- ϭ struct the time series corresponding to several d 6) but no seasonality have larger values of Vc. The importance of ENSO is examined by incorporat- EOFs, the associated PCs are combined. This ing the ENSO index into the simplest seasonal model reconstruction process preserves the phase of Downloaded from ϭ the time series. In addition, no information is (d 1) at different time lags ( f between 0 and 12 months). The smallest V value is obtained for the lost in the reconstruction, because the sum of all the individual reconstructed components gives ϭ c the original time series. Here the four first components were used, which explain 46 and 76% of model with f 11. This model is also selected over the seasonal one with an equal number of indepen- the variance for the cholera and ENSO time series, respectively. In (B), spectra were computed dent variables (d ϭ 2). It has a higher r2 value than with the maximum entropy method (MEM) applied to the reconstructed components in (A), the seasonal and autonomous models, accounting respectively. The dominant peak in both spectra corresponds to a frequency of 0.0225 months for a larger fraction of the variance. (For all models, (or 1/3.7 years). Results are robust to varying resolutions and MEM orders. Similar results were the number of neurons k used in fitting the function obtained with the multi-taper method (MTM). The peak corresponding to 0.0225 months was f of Eq. 2 varied between 1 and 5. Only the model significant at a 99% level and proved robust to varying resolutions and tapers. with the smallest Vc is reported here.) A value of k larger than one in the selected ENSO model indi- cates that a linear time series model is not adequate Fig. 3. The (square- to fit the causal relationships and would give a less root–transformed) chol- reliable conclusion about the role of ENSO. era data (black line) and the 2-months- ahead prediction of the Environ- Sea- GCV r 2 fitted model incorpo- mental sonal dkscore (%) rating both seasonality covariate clock Vc ϭ and ENSO at a lag f 11 (red line). – ϩ 1 2 0.937 55 – ϩ 2 2 0.932 57 – Ϫ 4 3 1.259 51 – Ϫ 6 3 1.176 60 ϩ EtϪ11 1 3 0.819 67
www.sciencemag.org SCIENCE VOL 289 8 SEPTEMBER 2000 1767 R EPORTS rule out overcompensation for disease levels incidence in the spring (from February into found for negative lags; that is, for the climate higher than the ones observed, this difference April). This inverse relation is indeed observed variables leading cholera by 4 to 6 months can result from a less effective immunity in the in the data. These results indicate that the dy- (Fig. 4). These maps also show coherence case of cholera. namics of cholera in Bangladesh are consistent with those obtained from correlations of the A positive effect of ENSO is observed on with a remote forcing by ENSO. climate anomaly fields with an ENSO index, the 2-month-ahead predictions of cholera inci- ENSO is thought to affect the atmospheric instead of cholera, and for a positive lag of 6 dence in the fall (November, for example). This circulation in the Indian Ocean and south Asia: months [(21) and figure 3 in (17)]. Thus, the is the time of the year during which the second Changes in cloud cover and evaporation associ- same changes in the atmospheric circulation and largest outbreak of cholera typically occurs ated with a weakening of the local Hadley cell in south Asia that trail the warming of the in the bimodal seasonal cycle. Cholera inci- increase the heat flux entering this region a few Pacific appear to anticipate changes in the dence 2 months ahead increases with both SST months after warming of the Pacific during an interannual variation of cholera in Bang- anomaly and present incidence. The sensitivity El Nin˜o (17). This provides a possible mecha- ladesh. The time lags in these associations are to changes in the ENSO index is largest for nistic connection between El Nin˜o and regional consistent with the 11-month delay found be- increasing negative values of the SST anomaly; climate variables potentially having an impact tween cholera and ENSO in our time series that is, for changes that anticipate an El Nin˜o, on cholera in Bangladesh. We considered three analysis. and for increasing and large positive values— interrelated climate variables: upper-tropospher- The above results suggest that an increase changes that occur during an El Nin˜o. Later in ic humidity, cloud cover, and top-of-atmosphere in local temperature ultimately mediates the the spring (April, for example), the levels of absorbed solar radiation. From global satellite– influence of ENSO on cholera. Higher ambi- cholera incidence are typically lower. Disease retrieved fields for these variables (18–20), we ent temperatures would correspond to higher levels can either increase or decrease with the computed global correlation maps between water temperatures in shallow bodies of wa- ENSO index. Incidence goes up with increasing the cholera time series in Dhaka and the cli- ter, such as ponds and rivers in the large negative values of the SST anomaly, as for mate time series at the different points of the estuary of Bangladesh and shallow coastal changes that would anticipate an El Nin˜o. It global grid, with temporal lags ranging from waters of the Bay of Bengal. Satellite data decreases, however, with increasing positive Ϯ12 months. The strongest associations are show that a positive association between values of the SST anomaly. One possible inter- pretation of this pattern is that large peaks in the fall are typically followed by small increases of on March 8, 2007 www.sciencemag.org
Fig. 5. Maps of correlation coefficients between cholera in Dhaka and temperatures on land and at sea for different lags, with the environmental variable anticipating disease. Monthly temperature data for the period from 1980 to 1995 were extracted from the Global Ocean Surface Temperature Atlas (GOSTAplus) for a grid of 5° latitude and longitude. Data were provided by the NASA Physical Oceanography Distributed Active Archive Center at the Jet Propulsion Laboratory/California In- stitute of Technology. Black boundaries indicate regions where the correlations are significant at the 0.002 level (see the legend to Fig. 4 for details). Downloaded from
Table 2. Parametric bootstrap test for the significance of additional predictor variables. Comparisons are made only across the best models previously selected with the GVC criterion. The models with previous disease levels and seasonality as the predictor variables are labeled as “dynamic seasonal,” and the model ϭ that in addition incorporates the ENSO index is labeled as “seasonal ENSO” ( f 11). The mean, SD, and ⌬ 2 ϭ maximum (Max.) value of i r are computed from the bootstrap data with n 1000. We assess first the significance of adding the ENSO index as a predictor variable by considering the dynamic seasonal models as the null hypothesis. In all cases, the addition of ENSO as a predictor variable is highly significant (␣ϭ 0.001). We then assess the significance of considering previous disease levels as a predictor variable. We compare the seasonal ENSO model, which does incorporate previous disease levels, to the corresponding ϭ model built only from the seasonal clock and the ENSO index, which we label “environmental” ( f 11, Fig. 4. Maps of correlation coefficients between k ϭ 3) . The environmental model performs poorly, with r 2 ϭ 44% (even for an equal number of neurons cholera in Dhaka and anomalies in two satel- k). The addition of previous disease levels as a predictor variable is highly significant (␣ϭ0.001). For all lite-retrieved fields 4 months earlier: (A) upper comparisons, we also recorded the number of times that the cross-validation criterion failed to select the ⌬ tropospheric humidity and (B) top-of-atmo- correct model (the null model). This frequency is given as the probability of iVc being negative in the last sphere absorbed solar radiation. The colored column. Results show that this frequency is extremely small for all comparisons. regions indicate areas of positive and negative correlations, and the black boundaries indicate Alternative Observed Mean SD Max. Prob. Null model the areas where correlations are significant at model ⌬ r 2 ⌬ r 2 ⌬ r 2 ⌬ r 2 ⌬ V Ͻ 0 the 0.002 level. (For each particular point in the 0 i i i i c map, the significance was calculated with a Dynamic seasonal (d ϭ 1) Seasonal ENSO 12.03 4.90 1.12 10.30 0.03 Monte Carlo test by randomly rearranging the Dynamic seasonal (d ϭ 2) Seasonal ENSO 9.93 2.00 1.77 7.61 0.024 elements of each time series at all points in 999 Environmental Seasonal ENSO 23.37 1.47 1.18 6.46 0.069 permutations.)
1768 8 SEPTEMBER 2000 VOL 289 SCIENCE www.sciencemag.org R EPORTS cholera and temperature is first observed to where p is the number of fitted parameters, and n is 16. See, for example, B. Finkensta¨dt, M. Keeling, B. T. Grenfell, Proc. R. Soc. London Ser. B 265, 753 (1998). the north of Bangladesh over the Himalayas, the sample size. V1 is the standard GCV criterion [see where temperature leads cholera increases by G. Wahba, Spline Models for Observational Data (So- 17. S. A. Klein, B. J. Soden, N. Lau, J. Clim.12, 917 (1999). ciety for Industrial and Applied Mathematics, Phila- 18. Monthly measurements of upper-tropospheric hu- 6 months (Fig. 5). The pattern then moves delphia, 1990)]. We used c ϭ 2 based on (11). This midity for the period 1979–92 [J. Schmetz, L. south, though it weakens, as the lag to cholera slight over-penalization of model complexity creates Vanderberg, C. Geijo, K. Holmlund, Adv. Space Res. decreases. Ambient temperatures have also a small bias toward simpler models but greatly re- 16, 69 (1995); B. J. Soden and F. P. Bretherton, J. Geophys. Res. 101, 9333 (1996)]. been implicated in the dynamics of diarrhoeal duces the chances of spuriously selecting an overly complex model. The FNN models were fitted with 19. Monthly measurements of cloud cover for the period diseases and of V. cholerae in the environ- FUNFITS, a suite of S/Fortran functions that run in 1983–91 are from the International Satellite Cloud ment in Peru (5, 22), and SSTs have been S-Plus (see www.cgd.ucar.edu/stats/Funfits/). Climatology Project [W. B. Rossow and R. A. Schiffer, Bull. Am. Meteorol. Soc. 72, 2 (1991)]. shown to display a bimodal seasonal cycle 15. We evaluate the significance of the improvement in fit between a “full” model that incorporates a pre- 20. Monthly measurements of top-of-atmosphere ab- similar to that of cholera cases in Bangladesh dictor variable and a “reduced” model that omits the sorbed solar radiation for the period 1985–89 are (2, 4). variable. The bootstrap test procedure consists of from the Earth Radiation Budget Experiment [E. F. Another mediating factor in the ENSO- generating a large number of artificial time series Harrison et al., J. Geophys. Res. 95, 18687 (1990)]. with the reduced model and fitting each of these 21. A supplementary figure is available at Science Online cholera relation might be the melting of the time series with both the full and reduced models. at www.sciencemag.org/feature/data/1051490.shl. snowpack in the Himalayas, through its effect The artificial time series are generated from the 22. A. A. Franco et al., Am. J. Epidemiol. 146, 1067 on the monsoons, precipitation, and river dis- reduced model by adding a vector of randomized (1997). charge. This scenario, which remains to be residuals to the vector of predictions from the re- 23. M. Ghil and K. Mo, J. Atmos. Sci. 48, 752 (1991). duced model. In the few cases where the resulting 24. , J. Atmos. Sci. 48, 780 (1991). investigated, is suggested by the strong but values are negative, we replace them by a lower 25. We thank K. Siddique and G. Fuchs for assistance reduced pattern appearing to the north of threshold of 0.1 (equal to the minimum value ob- with the cholera data; R. B. Sack, J. Trantj, and the Bangladesh (Fig. 5, first and second panels). served in the data). The improvement in fit between Office of Global Programs at the National Oceanic the full and reduced models on the original data is and Atmospheric Administration for stimulating this Floods and droughts can affect not only hu- compared to the improvements in fit on the artificial work; B. Soden for the cloud cover and radiation data; man interactions with water resources and time series, in which any apparent improvement is an and M. A. Rodriguez-Arias for computing assistance. artifact of the larger number of parameters and M.P. was supported by a James S. McDonnell Foun- therefore exposure to the pathogen, but also ⌬ 2 variables in the full model. Let i r denote the dation Centennial Fellowship and by The Knut and sanitary conditions and susceptibility to disease. difference in r2 between the full and reduced models Alice Wallenbergs Foundation; S.P.E. was supported for the ith time series (with i ϭ 0 being the original by a grant from the Mellon Foundation to S.P.E. and data and i ϭ 1,2,...n being the artificial data). Let N.G. Hairston Jr.; X.R. received partial support from p be the fraction of ⌬ r2, i Ͼ 0 values that are larger the Commissionat per Universitats i Recerca. References and Notes i on March 8, 2007 than ⌬ r2. The reduced model is then rejected in 1. For example, see J. L. Bryden, Epidemic Cholera in the 0 ␣ Ͻ␣ Bengal Presidency (Office of the Superintendent of favor of the full model at significance level if p . 19 April 2000; accepted 6 July 2000 Government Printing, Calcutta, India, 1871). 2. R. R. Colwell, Science 274, 2025 (1996). 3. P. R. Epstein, T. E. Ford, R. R. Colwell, Lancet 342, 1216 (1993). 4. B. Lobitz et al., Proc. Natl. Acad. Sci. U.S.A. 97, 1438 Myotonic Dystrophy in (2000). 5. W. Checkley et al., Lancet 355, 442 (2000). 6. E. Salazar-Lindo, P. Pinell-Salles, A. Maruy, E. Chea- Transgenic Mice Expressing an Woo, Lancet 350, 1597 (1997). 7. D. S. Broomhead and G. P. King, Physica D 20, 217 www.sciencemag.org (1986). Expanded CUG Repeat 8. R. Vautard and M. Ghil, Physica D 35, 395 (1989). 1 1 2 1 9. Examples of such responses to seasonal forcing in Ami Mankodi, Eric Logigian, Linda Callahan, Carolyn McClain, nonlinear models for disease dynamics can be found Robert White,1 Don Henderson,1 Matt Krym,1 in W. M. Schaffer et al., in The Ubiquity of Chaos,S. 1 Krasner, Ed. (American Association for the Advance- Charles A. Thornton * ment of Science, Washington, DC, 1990), pp. 138– 166; I. B. Schwartz and H. L. Smith, J. Math. Biol. 18, Myotonic dystrophy (DM), the most common form of muscular dystrophy in 233 (1983); and I. B. Schwartz, J. Math. Biol. 30, 473 adult humans, results from expansion of a CTG repeat in the 3Ј untranslated Downloaded from (1992). region of the DMPK gene. The mutant DMPK messenger RNA (mRNA) contains 10. S. Ellner and P. Turchin, Am. Nat. 145, 343 (1995). 11. D. W. Nychka, S. Ellner, A. R. Gallant, D. McCaffrey, an expanded CUG repeat and is retained in the nucleus. We have expressed an J. R. Stat. Soc. B 54 399 (1992). untranslated CUG repeat in an unrelated mRNA in transgenic mice. Mice that 12. F. Takens, in Dynamical Systems and Turbulence,D. expressed expanded CUG repeats developed myotonia and myopathy, whereas Rand and L. S. Young, Eds., Lecture Notes in Mathe- matics, vol. 898 (Springer-Verlag, New York, 1981), mice expressing a nonexpanded repeat did not. Thus, transcripts with expanded pp. 366–381. CUG repeats are sufficient to generate a DM phenotype. This result supports 13. M. Casdagli, in Nonlinear Modeling and Forecasting, a role for RNA gain of function in disease pathogenesis. M. Casdagli and S. Eubank, Eds. (Addison-Wesley, New York, 1992). Myotonic dystrophy (DM, prevalence 1 in 7400 sults from the expansion of a CTG repeat in the 14. To fit f we used the feedforward neural network Ј Ј (FNN) model live births) is characterized by dominantly in- 3 untranslated region (3 UTR) of the DMPK ͑ ͒ ϭ  herited muscle hyperexcitability (myotonia), gene, which encodes a serine-threonine protein f x1, x2, ..., xd 0
k d progressive myopathy, cataracts, defects of car- kinase (2). The transcripts from the mutant ϩ  ␥ ϩ diac conduction, neuropsychiatric impairment, allele are retained in the nucleus (3, 4), and iG ijxj i (2) i ϭ 1 j ϭ 1 and other developmental and degenerative levels of DMPK protein are correspondingly where G is a sigmoid function such as G( y) ϭ e y/ manifestations (1). This complex phenotype re- reduced (5). The expanded repeat also changes (1 ϩ e y). Given k and the set of independent vari- the structure of adjacent chromatin (6) and  ␥ ables (x1, x2,...xd), the model parameters ( i, ij, i) silences the expression of a flanking gene (7, 8), 1 2 were estimated by ordinary least squares. Models Department of Neurology, Department of Neurobi- SIX5, which encodes a transcription factor. with different values of k or a different set of x’s were ology and Anatomy, School of Medicine and Dentist- compared with a GCV criterion function ry, University of Rochester, Box 673, 601 Elmwood The effects on DMPK and SIX5 expression RMS 2 Avenue, Rochester, NY 14642, USA. may account for particular aspects of the DM V ϭ (3) c c phenotype. Dmpk knockout mice have reduced Ϫ *To whom correspondence should be addressed. E- 1 p n mail: [email protected] force generation in skeletal muscle (9) and ab-
www.sciencemag.org SCIENCE VOL 289 8 SEPTEMBER 2000 1769